UsingVoteCounts’DigitstoDiagnoseStrategiesand Frauds: Russiawmebane/apsa13.pdfOver the most recent election cycles Russian elections have become increasingly unfree and unfair, characterized

Using Vote Counts’ Digits to Diagnose Strategies and

Frauds: Russia∗

Walter R. Mebane, Jr.†

August 11, 2013

∗Prepared for presentation at the 2013 Annual Meeting of the American Political Science Asso-ciation, Chicago, August 29–September 1, 2013.

†Professor, Department of Political Science and Department of Statistics, University ofMichigan, Haven Hall, Ann Arbor, MI 48109-1045 (E-mail: [email protected]).

Abstract

Tests of the digits of vote counts have been proposed to diagnose election fraud. Both the

second-digit Benford’s-like Law (2BL) and the idea that the last digits should be uniformly

distributed have been proposed as standards for clean elections. Many claim that election

fraud is rampant in recent Russian federal elections (since 2004), so Russia should be a

good setting in which to see whether the digit tests add any diagnostic power. Using

precinct-level data from Russia, I first use a randomization test to identify sets of precincts

(called UIKs in Russia) in which vote counts for candidates are augmented compared to

vote counts in a comparison sets of UIKs. These are a subset of UIKs in which turnout

percentages or the percentage of votes for Putin (or United Russia) are divisible by five.

Then I run tests of the second and last digits of the UIK vote counts both for the entire set

of UIKs in an election year and separately for various sets of UIKs. The digit tests produce

surprising and on balance implausible results. For example, they suggest that none of the

votes for Putin in 2004 and 2012 or for United Russia in 2011 were fraudulent, while votes

for Medvedev in 2008 were fraudulent. The usefulness of simple and direct application of

either kind of digit tests for fraud detection seems questionable, although in connection

with more nuanced interpretations they may be useful.

1 Introduction

In authoritarian regimes in recent decades election fraud and its detection have been a

concern (Lehoucq 2003; Bjornlund 2004; Schedler 2006; Alvarez, Hall and Hyde 2008;

Myagkov, Ordeshook and Shaikin 2009; Cantu and Saiegh 2011; Fukumoto and Horiuchi

2011; Simpser 2013). Over the most recent election cycles Russian elections have become

increasingly unfree and unfair, characterized by suppression of electoral competition, rising

levels of administrative interference and drastic growth of electoral frauds (Freedom House

2010). The credibility of the elections has fallen to such an extent that one observer

described them as “election type events” rather than genuine elections (Gel’man 2009).

The growth of electoral manipulations and crude falsifications, and their widespread

systematic pattern has been described as “mass administrational electoral technology”

(Buzin and Lubarev 2008; Lubarev, Buzin and Kynev 2007). Evidence of Russian election

frauds is found by inspecting vote distributions, which are distinctly bimodal and spiky

(Shpilkin 2011; Klimek, Yegorov, Hanel and Thurner 2012), or by conducting field

experiments (Enikolopov, Korovkin, Petrova, Sonin and Zakharov 2013). Fraudulent voter

turnout is readily detected (Myagkov and Ordeshook 2008; Myagkov, Ordeshook and

Shaikin 2008, 2009; Mebane and Kalinin 2009b,a). Ballot box stuffing and simply writing

down false numbers are likely mechanisms (e.g. Boldyrev 2012), but phony voter

registrations (Arbatskaya 2004) also occur.

Kalinin and Mebane (2011) argue that recent Russian electoral frauds may be partly

explained by institutional incentives connected to recentralization. To address deficits in

the Russian party system (Golosov 2004) and “the problem of dysfunctional bargaining”

(Filippov, Ordeshook and Shvetsova 2004, 174), United Russia becomes “an integrated

party that facilitates cooperation across levels of government” (Filippov, Ordeshook and

Shvetsova 2004, 195) at the price of some “perversion of democracy” (Filippov, Ordeshook

and Shvetsova 2004, 174). Kalinin and Mebane (2011) argue that the recentralization

involves coordination among officials using particular values in election returns to

1

implement a scheme of “signaling” between regional actors and the center. In particular

Kalinin and Mebane (2011) claim that turnout values that are divisible by five, which as

Mebane and Kalinin (2009b,a) show occur especially often, are used by regional and more

local actors to signal loyalty to the Kremlin. The signaling is consequential for more than

election outcomes: Kalinin and Mebane (2011) show that these “signals” are connected to

systematic movements in transfer payments from the center to regional governments.

Evidence of election frauds in Russia is ample, so these elections provide a convenient

setting for examining whether tests of the digits in precinct vote counts have anything to

contribute to efforts to diagnose the manipulations the frauds involve. Notwithstanding the

skeptical arguments (Carter Center 2005; Shikano and Mack 2009; Lopez 2009; Deckert,

Myagkov and Ordeshook 2011; Mebane 2011, 2013), the 2BL tests based on the second

significant digits originally introduced by Pericchi and Torres (2004, 2011) are of interest.1

Also there is a suggestion by Beber and Scacco (2012) that vote counts’ last digits can be

diagnostic. Beber and Scacco (2012) argue that if there are no frauds then the last digits of

vote counts should be uniformly distributed. If fraud is as rampant in Russian elections as

many observers claim, then they should be easy marks for the digit tests, if the tests

actually work to detect fraud.

A caveat is that the motivation for the digit tests presumes that those who commit

election fraud are unsophisticated or careless, or that the mechanisms used to commit the

frauds do not allow precise control of what the fraudulent outcomes are. Beber and Scacco

(2012) argue that humans who fake vote counts simply by writing down numbers they

happen to think of are subject to psychological limitations that produce nonuniform

patterns in the results. Such human limitations would be easily overcome, say by using a

random number generator to create the fake numbers: using well-known algorithms, the

fake numbers can have any desired distribution. Vote counts that are 2BL-distributed are

easy to simulate as well, which would tend to undercut the test advocated by Pericchi and

1Mebane (2006) introduces the location 2BL (for second-digit Benford’s Law) tests, even though it ismore precise (Mebane 2010) to use 2BL to refer to second-digit Benford-like tests.

2

Torres (2011). Indeed, Beber and Scacco (2012) point out that the simulated counts

produced by one of the mechanisms in Mebane (2006) that was designed to produce counts

that satisfy 2BL also have uniformly distributed last digits. Given that officials committing

election fraud in Russia would know that precinct data would be released and that many

would be scrutinizing the data, it is reasonable to believe that any such official might

anticipate tests that might be performed.

I check whether digit tests are triggered by the votes cast in places that by other

evidence feature many artificially manipulated and likely fraudulent votes. First I isolate

UIKs2 in which unusually large numbers of votes are counted for candidates or parties

contesting the election. Then I compare digit test statistics for data from those UIKs to

statistics for data from the other UIKs. Perhaps fraud is so pervasive in Russian elections

that a UIK having an unusually large ballot count does not actually distinguish how much

the votes counted there have been manipulated. But the effort to isolate places where

fraud is more prevalent is at least plausible.

2 Randomization, etc.

For each UIK, turnout3 or the proportion of votes for Putin (or, depending on the year, for

Medvedev, for United Russia or for a candidate affiliated with United Russia) is multiplied

by 100. Let Ti denote this value of turnout in UIK i, and let Si denote the value of the vote

percentage. Let Vik denote the number of ballots counted for alternative k in UIK i. Using

the indicator function I(·),4 for each of the divisible-by-five values j = 0, 5, . . . , 100 let

VTjk =

∑i I(|Ti − j| < .5)Vik∑i I(|Ti − j| < .5)

(1)

2A UIK, uchastkovaya izbiratelnaya komissiya, is a precinct.3Turnout is the sum of the number of ballots issued to voters who voted early, the number of ballots

issued to the polling station on election day and the number of ballots issued outside the polling station onelection day divided by the number of voters included in the voters list.

4I(·) = 1 if the argument condition is true, I(·) = 0 otherwise.

3

denote the mean number of ballots counted for alternative k in UIKs whose rounded value

of Ti would be j. For each j = 0, 5, . . . , 100 let Tjk = {VTjk − Vik : .5 ≤ |Ti − j| < 2.5}

denote the set of differences between VTjk and Vik for UIKs that have Ti within 2.5 percent

of the divisible-by-five value j but for which Ti would not round to j. Likewise based on

VSjk =

∑i I(|Si − j| < .5)Vik∑i I(|Si − j| < .5)

, (2)

define Sjk = {VSjk − Vik : .5 ≤ |Si − j| < 2.5} relative to vote percentages.

The values in Tjk and Sjk can be used to perform tests for ballot box stuffing (or other

fakery that has the same effect on vote totals) that is tied to signaling among officials as

described by Kalinin and Mebane (2011). The values in Tjk and Sjk represent comparisons

of UIKs that have a divisible-by-five turnout figure or United Russia vote percentage to

other UIKs that are similar in that they have similar turnout or vote percentage values.

While in general these UIKs are quite diverse, notwithstanding their heterogeneity they

ended up having similar turnout or vote percentages. They are in this sense “neighbors.”

So we might expect the ballot counts for each alternative k to be likewise similar among

these “neighboring” UIKs. If UIKs with one of the divisible-by-five values for rounded Ti or

rounded Si do not have artificially elevated counts of ballots for alternative k, then the

typical value of the elements of Tjk and Sjk should be zero. On the other hand, if some of

the divisible-by-five turnout values are used to implement signaling among officials, as

described by Kalinin and Mebane (2011), then we might expect the elements of Tjk to be

typically positive for some values of j. While Kalinin and Mebane (2011) focus on signaling

via turnout, analogous considerations suggest there may be signaling using vote

percentages directly, hence such signaling may imply that the elements of Sjk are typically

positive for some values of j.

One way to implement such tests with minimal assumptions is to use a randomization

test procedure. To do this for turnout and a divisible-by-five number j, I consider each set

4

of UIKs that have rounded Ti equal to j together with the “neighbors” of these UIKs: let

Ij = (i : I(|Ti − j| < 2.5)) denote an ordered set of the indices of these UIKs. For each

permutation σ(Ij) of Ij , compute

V σTjk =

∑i I(|Ti − j| < .5)Vσ(i)k∑

i I(|Ti − j| < .5)(3)

and T σjk = {V σ

Tjk − Vσ(i)k : .5 ≤ |Ti − j| < 2.5}. To test whether the mean Tjk of the

elements of Tjk is significantly large given the hypothesis that there is nothing special

about the UIKs that have rounded Ti equal to j compared to the “neighbors” of these

UIKs, find a p-value by computing the proportion of values T σjk among the permuted means

T σjk that are greater than Tjk. That proportion is the p-value.

The procedure for vote percentages, based on Sjk, VσSjk and Sσ

jk, is exactly analogous.

I use Tjk and Sjk with data for each Russian presidential election during 2004, 2008 and

2012 and for each Duma election during 2003, 2007 and 2011 to run randomization tests of

whether UIKs with rounded turnout or United Russia vote percentages that are divisible

by five have excess ballots.5 In 2007 and 2011 the elections used proportional

representation electoral rules, but in 2003 half the seats were allocated based on

proportional representation while half were determined by plurality rule elections in

single-member districts. In 2003 I consider the proportional and plurality results

separately. Because the number of UIKs is large (greater than 95,000), the number of

elements in each set Tjk or Sjk is too large to allow all permutations to be enumerated. I

use a Monte Carlo method based on using 10,000 random permutations to determine each

p-value. Because the test for each year’s votes involves multiple comparisons, I use methods

based on the FDR (Benjamini and Hochberg 1995) to adjust the test level in each year for

the set of divisible-by-five values j and choice alternatives k that occur in each year.6

5UIK-level data are downloaded from sites connected to Central Election Commission of the RussianFederation (2013).

6To define FDR adjustment for J tests, sort the p-values with p(j) being the j-th ordered value (p(1) issmallest). For test level α, let d be the smallest value such that p(d+1) > (d+1)α/J . The d smallest p-valuesmatch tests rejected after FDR adjustment (Benjamini and Hochberg 1995) .

5

The test results suggest that in the presidential election of 2012 there is significant

artificial vote count augmentation that focuses on turnout values that are divisible by five.

Figure 1 shows test results for all five candidates that relate to turnout (Tjk). Tests are

attempted for the values j = 20, 25, . . . , 100.7 Several p-values in Figure 1 are less than the

conventional test level α = .05, and several are significant at level α = .05 even given FDR

adjustment for multiple testing. For Zhirinovskij, Mironov and Prohorov, the significant

p-values occur for j = 100, while for Putin there is a significant p-value for j = 85. Other

p-values that are not significant given multiple testing but are nonetheless small (less than

.1) occur for Putin at j ∈ {45, 90}, for Zyuganov and Mironov at j = 50, for Prohorov at

j = 60 and for Zyuganov, Mironov, Prohorov and Putin at j = 80.

*** Figure 1 about here ***

The test results suggest even more that in 2012 vote counts are artificially augmented

when Putin has a vote percentage value that is divisible by five. In Figure 2, which shows

results based on Sjk, significant (FDR adjusted) p-values occur for Zhirinovskij, Zyuganov,

Mironov, and Putin at j ∈ {70, 75, 80, 85}, for Zyuganov at j = 90 and for Putin at

j = 100. Small p-values also occur for Zyuganov at j ∈ {30, 65}, for Mironov at j = 30, for

Prohorov at j ∈ {30, 35, 70, 75} and for Putin at j ∈ {30, 65, 90}. In UIKs at many

divisible-by-five values of Putin’s vote percentage greater than or equal to 65 percent and

at a couple of lower percentages, the counts of ballots for most or all of the candidates are

typically greater than in “neighboring” UIKs.


How often are UIKs that have turnout divisible by five the same as the UIKs that have

Putin’s vote percentage divisible by five? Figure 3, which displays how often each

combination of rounded turnout and rounded Putin vote percentage values occur in the

7The absence of p-values for j = 20 in Figure 1 results from there being no UIKs matching that value.p-values are missing for some values of j in Figures 7, 15 and 9 for the same reason.

6

2012 UIK data, provides some of the answer. The peak occuring with a frequency of 533

where both percentages are at 100 is the most notable feature of the graph. Somewhat

harder to see are the local peaks that occur at combinations of values that are divisible by

five. For instance there are local peaks at turnout equal 100 percent and Putin percentage

equal to 50, 55, 60, 65 or 70. Other local peaks at divisible-by-five values are harder to pin

down in Figure 3 but are clear upon inspecting the counts in each cell. For turnout greater

than 50 and Putin percentage greater than 55, counts in twenty divisible-by-five cells are

greater than in any of the “neighboring” cells (i.e., one or two units away). The list of

these cells appears in Table 1.8

*** Figures 3 and Table 1 about here ***

Figure 4, which shows the smoothed joint density of the UIK-level turnout and Putin

vote proportion variables, reinforces the message that divisible-by-five values of turnout

and of Putin vote percentage occur occur relatively frequently for the higher values of each

variable.9 The density plot does not resolve the location of the local maxima as precisely as

does inspecting the counts in the frequency distribution of the rounded variables, but it

does clearly show the regularity of the pattern in which many of the local peaks occur.

Peaks are apparent near turnout of .5, .7, .85 or 1.0 and Putin vote proportion of .55, .75,

.85 or 1.0.


If not for the evidence of the randomization tests reported in Figures 1 and 2, we might

explain the greater frequency of turnout and Putin vote percentage figures that are divisible

by five as merely due to officials at some level rounding off reported figures. Regarding this

8Other divisible-by-five cells occur more frequently than cells one unit away but not for all cells that aretwo units away: the referent (turnout, vote percentage) pairs are (80, 75), (85, 65), (90, 65), (90, 95), (95, 75)and (95, 95).

9The density is computed using the sm package of R (R Development Core Team 2005). I call functionsm.density() with argument h=c(.002,.002), which produces a window for the normal kernel functionthat is narrower than optimal. With optimal smoothing parameters the density resembles Figure 3.

7

potential explanation, however, note that such rounding off would be occurring in

computed percentages and not in the original counts of who voted and for whom. The

randomization test results show that the UIK counts of votes received by the candidates in

divisible-by-five UIKs are augmented in comparison to the “neighbors” of those UIKs.

Rather than rounding off after the fact, one suspects that the divisible-by-five values came

first and the vote counts were then concocted to go with the specified percentages.

The relationship between turnout and the leading candidate’s vote proportion in the

presidential elections of 2008 and 2004 is similar to the relationship in 2012. The plots of

the smoothed densities for 2008 and 2004, shown in Figures 5 and 6, resemble the 2012

density in that the local peaks are in the same locations. The peaks when turnout and the

vote proportion (for Medvedev in 2008 and for Putin in 2004) both equal 1.0 are higher in

2008 and in 2004 than the peak for the same values in 2012. The peaks near .75 and .85 for

the candidates’ vote proportion seem to be roughly the same size in all three years, while

the peaks near .55 are higher in 2012 than in the other two years.

*** Figures 5 and 6 about here ***

Randomization tests using the earlier presidential election data produce results similar

to those seen for 2012, but also there is evidence that the amount of artificial vote count

augmentation tied to the divisible-by-five values increases over time. In 2008 the frequency

of focused vote augmentation across distinct divisible-by-five values seems roughly

comparable to the frequency observed in 2012. In 2008 all the significant p-values for

divisible-by-five turnout values occur for vote counts for Medvedev (Figure 7(a)): at

rounded turnout j ∈ {80, 90, 95}. Small p-values occur for all candidates at j = 65, for

Medvedev and Zhirinovskij at j ∈ {75, 85}, and for Zhirinovskij at j ∈ {80, 90}. Tests

focused on instances when Medvedev has a divisible-by-five vote percentage also show

significant results about as frequently as in 2012. There are significant p-values for

Medvedev at j ∈ {65, 70, 75, 80, 85}, for Zhirinovskij at j ∈ {70, 75, 80, 85, 90}, for

8

Zyuganov at j ∈ {65, 70, 85} and for Bogdanov at j ∈ {80, 85} (Figure 8). Several small

p-values also occur for all candidates for j ≥ 60.


In 2004 the number of distinct divisible-by-five values that test positive for vote

augmentation is slightly smaller than in the subsequent two presidential election years.

Only one p-value is significant among the p-values for turnout, for the Against All

alternative at j = 100 (Figure 9(g)). Many p-values other than that one are small at j ≥ 55,

however, and vote counts for all candidates (and for Against All) are involved at various

values of j. In this respect, the number of j values at which votes are augmented may be

somewhat greater in 2004 than in 2012. Augmentation appears less prevalent in 2004

among the divisible-by-five values of Putin’s vote percentage. A single p-value is significant

for each of Putin, Haritonv and Malyshkin, at j = 95 (Figure 10). Several p-values are

small for many candidates, at j ∈ {45, 85, 90, 95}, but the small p-values occur for fewer

distinct value of j than in 2012 or 2008. In this sense the prevalence or perhaps it’s better

to say the diversity of artificially augmented vote counts appears to increase over time.


The pattern and amount of artificial vote augmentation in the 2011 and 2007 Duma

elections seem to be similar to what happens in the 2012 and 2008 presidential elections.

The relationship between turnout and United Russia’s vote proportion in these Duma

elections resembles the relationship between turnout and the leading candidate’s vote

proportion in the presidential elections. The smoothed density for 2011 has a dominant

peak at turnout and United Russia proportion both equal to 1.0, and other local peaks

occur when turnout is near .45, .6 or .8 and the United Russia proportion is near .4, .6 or

.8, when turnout equals 1.0 or when the United Russia proportion equals .2 (Figure 11).

The pattern in the smoothed density for 2007 is similar: local peaks that are apparent in

9

2011 when the United Russia proportion equals .2 are not present in 2007, and local peaks

occur when turnout is near .5, .65 or .8 and the United Russia proportion is near .5, .6 or

.8 (Figure 12). Neither the pairs in the set {(i, j) : i = 45, 60, 80, j = 40, 60, 80} in 2011 nor

the pairs in the set {(i, j) : i = 50, 65, 80, j = 50, 60, 80} in 2007 are always larger than all

their “neighbors” in the joint distribution of the rounded turnout and United Russia

percentage variables, but those values are “neighbors” of other local modes.


The set of divisible-by-five values on which randomization tests suggest vote

augmentation is concentrated is similar between the Duma and presidential election years,

and again there is evidence of a progression over time. In 2011, as in 2012, among the

turnout-based tests a single p-value is significant for all parties except KPRF and United

Russia, at j = 100 (Figure 13). For United Russia in 2011, as for Putin in 2012, the

significant p-values occur at slightly lower values, namely j ∈ {75, 85}. For the tests based

on vote percentage, significant p-values occur about as often and for about the same values

of j in both 2011 and 2012 for the parties that competed in both elections (Figure 14). In

2007 the turnout-based tests resemble the pattern in 2012 in having significant p-values at

j = 100 for several small parties and a significant p-value for United Russia at a slightly

smaller value, namely j = 95 (Figure 15). A few other p-values are also significant, and

each party has at least a few small p-values. The tests based on vote percentage in 2007

feature slightly fewer significant p-values than in 2012, as well as slightly fewer small

p-values for parties that competed in both elections (Figure 16).

*** Figures 13, 14, 15 and 16 about here ***

The pattern of votes being significantly augmented at divisible-by-five turnout and

United Russia vote percentage values becomes more prevalent over time at least in the

sense that significant augmentations are associated with more distinct values of those two

10

variables over time. Moreover given the very different patterns observed in the 2003 Duma

election, it is reasonable to say that the divisible-by-five patterning in vote augmentations

began in the 2004 presidential election.

Both the distribution of divisible-by-five turnout and United Russia vote percentage

values and the randomization test results differ in 2003 from the findings from subsequent

years. The most immediate difference in the smoothed density for the proportional

representation votes is that in 2003 the peak when both turnout and United Russia vote

proportion equal 1.0 is no longer the highest mode: a mode near the turnout-proportion

pair (.55, .35) is higher (Figure 17). Inspecting the joint distribution of rounded turnout

and United Russia vote percentage values shows the greatest count, which is 161, occurs

for (56, 34), while the count for (55, 35) is only 121. In that joint distribution, the only

divisible-by-five pairs that are local peaks among their “neighbors” are (75, 60), (95, 90)

and a few pairs in which turnout rounds to 100 percent.10 In fact the pair (100, 100) is not

a local peak—the count for (100, 99) is one greater (135 versus 134). This suggests there is

not substantial coordination on a few values that are divisible by five. The randomization

test based on turnout and proportional representation votes finds three significant p-values,

a result arguably not all that different from the pattern for turnout in 2012, but in the test

based on United Russia percentage no p-values are significant (Figures 18 and 19). All the

significant p-values in the turnout test are for turnout value j = 100 and affect votes for

United Russia, LDPR and Yabloko, so there is some evidence of artificial vote

augmentation. But the absence of significant results in the test based on United Russia

percentage imply that whatever is going on in this election is unlike what happens in the

later years.

*** Figures 17, 18 and 19 about here ***

The votes cast in the plurality rule elections in 2003 show even fewer signs of vote

augmentation related to turnout or United Russia percentage values that are divisible by

10These pairs are (100, 50), (100, 60), (100, 80) and (100, 85).

11

five. The smoothed density of the two variables has many local peaks, but only eight of

these are centered on a pair of divisible-by-five values in the joint distribution of the

rounded variables (Figure 20).11 None of those pairs occur very frequently—the largest

count has the pair (55, 60) occurring in 28 UIKs—and the pair (100, 100) is not a local

peak. The randomization test based on turnout produces three significant p-values, but

none of these are for United Russia (Figure 21). The randomization test based on United

Russia percentage produces no significant p-values (Figure 22). While KPRF has a single

p-value that is less than .1, none of the p-values for United Russia or for LDPR are small.

If there is vote augmentation in these elections, it occurs in patterns utterly unlike what

happens in later years.

*** Figures 20, 21 and 22 about here ***

3 Digit Tests

Many have argued there is strong evidence of widespread election frauds in Russian federal

elections at least since 2004 (e.g. Myagkov, Ordeshook and Shaikin 2009; Klimek et al.

2012), and the tests discussed in the previous section have identified specific sets of UIKs in

which vote counts are systematically greater than the counts in “neighboring” UIKs. The

question for this section is whether any of the various tests that have been proposed that

examine digits in the vote counts respond to the alleged frauds.

In addition to tests based on the second significant digits examined previously, of

especial interest is the test proposed by Beber and Scacco (2012), which is based on the

idea that if there are no frauds then the last digits of vote counts should be uniformly

distributed. For the last digits to be uniformly distributed means that they should occur

equally frequently. This is analogous to the second-digit equal frequency (2E) hypothesis

considered in Mebane (2006), except the the expected frequency of 110

is applied to the last

11The pairs are (40, 50), (55, 60), (60, 75), (65, 15), (70, 30), (90, 50), (100, 40) and (100, 50).

12

digit of each vote count. To test whether the hypothsis that the last digits in a set of NL

vote counts is uniformly distributed, I let ℓi be the number of times the last digit is i in the

set and define the Pearson chi-squared test statistic X2LU =

∑9i=0(ℓi −NL/10)

2/(NL/10),

where NL =∑9

i=0 ℓi. To compare X2LU to the chi-squared distribution with nine degrees of

freedom, I use a significance probability of the form advocated by Pericchi and Torres

(2011), αL = (1 + [−epL log(pL)]−1)−1 where where pL is p-value of X2

LU .

Vote counts in UIK-level data may be small, and indeed Beber and Scacco (2012, 217)

warn against using their test with small counts. Indeed, with small counts an expectation

that the last digits should be uniformly distributed goes against the suggestion from

Pericchi and Torres (2011) that 2BL should describe the distribution of the

second-significant digits: if the counts are two-digit numbers then the last digit is the

second digit. To avoid including small counts I restrict attention to vote counts greater

than 99 by defining a modified statistic X2LU∗ =

∑9i=0(ℓi∗ −NL∗/10)

2/(NL∗/10), where ℓi∗ is

the number of times the last digit is i and the count is greater than 99 and NL∗ =∑9

i=0 ℓi∗,

with significance probability αL∗ = (1 + [−epL∗ log(pL∗)]−1)−1 where pL∗ is p-value of X2

LU∗.

If the last-digits of all the counts are in fact uniformly distributed, then removing the

counts with values between zero and 99 should not affect the test except by reducing its

power due to the smaller number of cases.

The randomization test results for 2004–2012 suggest it is reasonable to consider not

only digit tests for the entire set of UIKs in each election but also in subsets that roughly

match the apparent occurrence of vote augmentation. UIKs that have rounded turnout and

round Putin (or United Russia, etc.) vote percentages that are divisible by five should be

considered separately from the others, and Figures 2, 8 and 14 suggest further separating

the UIKs in which the Putin/United Russia percentage is greater than 60 from the rest. To

this end I compute digit test statistics for four sets of UIKs, symbolized and defined as

follows:

All the set of all UIKs;

13

D5>60 UIKs in which rounded turnout and rounded Putin vote percentage are divisible

by 5 and Putin vote percentage is greater than 60;

ND5>60 UIKs in which neither rounded turnout nor rounded Putin vote percentage

are divisible by 5 and rounded Putin vote percentage is greater than 60;

ND5<60 UIKs in which neither rounded turnout nor rounded Putin vote percentage

are divisible by 5 and rounded Putin vote percentage is less than or equal to 60.

Given the randomization tests, the UIKs in D5>60 likely have augmented vote counts, while

the UIKs in ND5<60 do not have any of the divisible-by-five features that may be used to

signal and also have relatively low precentages supporting Putin or United Russia. I use

ND5>60 to offer a contrast with D5>60 with regard to the divisible-by-five values and a

contrast with ND5<60 on the Putin/United Russia percentage. Clearly, based on the

signaling story and randomization test results, we should expect the vote counts for the

UIKs in D5>60 to show the clearest signs of being fraudulent. But obviously the other sets

may show signs of fraud if fraud is prevalent but the signaling story is wrong or if, the

randomization test results notwithstanding, vote augmentation is not the only device used

to commit fraud.

All this supposes that the digit tests detect fraud. Mebane (2013) and the discussion of

conditional second-digit tests already give ample reason to doubt that X22BL is effective for

that. Whether X2LU or X2

LU∗ work to detect fraud in Russia remains to be seen.

It also supposes that those committing any frauds in Russia are unsophisticated. In

fact, claims about election fraud were well known to officials in Russia, and there is every

reason to believe they were familiar with digit tests among others.12 One must credit the

possibility that any faked vote counts are designed to pass various tests. Digit tests would

be easy to pass with fake data if the fakery were sufficiently well coordinated. The

12Key election officials attended workshops held to discuss findings regarding Russian election frauds andattempted to defend the elections.

14

institutional requirements for signaling (Kalinin and Mebane 2011) are at least one reason

why other markers for fraud are nonetheless easy to detect.

The second-digit tests applied to data from the 2012 election show extensive deviations

from 2BL, but it is unclear whether these deviations should be interpreted as diagnosing

fraud. In the set of all UIKs, X22BL is significant for all candidates except Zyuganov (Table

2). j differs significantly from j for Zhirinovskij, Mironov and Prohorov but not for

Zyuganov or Putin.13 Remarkably, X22BL is not significant for any of the candidates in the

set D5>60, but X22BL is significant for all candidates in both ND5>60 and ND5<60. j for

Putin is significantly greater in ND5<60 than in ND5>60. In ND5>60 j < j while in ND5<60

j ≈ 4.35. Simply applying the interpretation suggested by Mebane (2010) would suggest

that Putin gained strategically switched votes in UIKs where he received a low proportion

of the vote, while such gains did not occur in UIKs where he received most of the votes.

Intuitively that seems like an odd result, but because no one expected Putin to lose the

election and because the other candidates all received nontrivial and unequal shares of the

vote,14 it is difficult to say what if any strategy most voters might have been using. And it

remains unclear to what extent the data are fraudulent. The most cynical interpretation,

perhaps, is that in the UIKs in D5>60, vote counts were substantially faked with an eye to

passing the test set out by Pericchi and Torres (2004, 2011), while in the UIKs in ND5<60

there was less fakery and the vote counts more reflect Putin’s actual support.

*** Table 2 about here ***

The last-digit test applied to the 2012 data produces eerie results that also appear in a

sense too good to be true. The least one can say is that the results are exceptionally good

for Putin. X2LU∗ is never significant for Putin in any of the subsets, although the statistic in

13j is the mean of the second digits. If the distribution of the counts’ second-digits has frequencies rj as

given by Benford’s Law, then the second digits’ expectation is j =∑9

j=0 jrj = 4.187.14Based on Central Election Commission of the Russian Federation (2013), Putin received 45,602,075 votes,

which is 64.4 percent of the valid votes, while the other candidates (Zhirinovskij, Zyuganov, Mironov andProhorov) respectively received 6.3, 17.4, 3.9 and 8.1 percent. This outcome does not match the equilibriaof, for example, Cox (1994).

15

the set of all UIKs comes close with αLU∗ = .06 (Table 3). Remarkable is that in all sets

X2LU < X2

LU∗ for Putin: even in the set of all UIKs, including the additional 12,367 UIKs

that have values between zero and 99 increases the fit to a uniform last-digit distribution.

X2LU∗ is not significant in D5>60 for any candidate except Zyuganov, X2

LU∗ is significant for

every candidate except Putin in ND5>60, and X2LU∗ is significant for every candidate except

Putin and Zyuganov in ND5<60.15


It is difficult to think of a natural electoral process in which there are no artificial

elements that would produce a result where some candidates have vote counts with

uniformly distributed last digits and others do not, and moreover some candidates have

vote counts with uniformly distributed last digits in some places but not others. So

following Beber and Scacco (2012), one might say the last-digit test diagnoses fraud. But

then, oddly, the test would suggest that all the candidates had fraudulent vote totals

except for Putin and occasionally except for Zyuganov, and except for all the candidates

(except now Zyuganov) in those UIKs where the randomization test shows vote counts

were augmented. Beber and Scacco (2012) describe psychological mechanisms to motivate

their test, but it is difficult to understand how such mechanisms could possibly explain the

patterns in Table 3. More likely is the cynical explanation that Putin’s vote counts are

generally faked everywhere—faked using sophisticated algorithms and coordination and not

merely in the UIKs in D5>60—and that the fakery’s effect on the vote counts for other

candidates is intricate. Those effects probably trace back to Putin’s relationships with the

other candidates and might well not be accidental.

The patterns in the digit tests in 2011 resemble 2012 in some ways, but there are some

key differences. The second-digit tests show that in the set of all UIKs X22BL is significant

for all parties except KPRF (Table 4). KPRF is Zyuganov’s party in 2012, and in the set

15The two p-values that equal .05 are not significant if FDR adjustment is applied to the set of all p-valuesin D5>60, ND5>60 and ND5<60.

16

of all UIKs Zyuganov’s vote counts were the exceptional case for X22BL in 2012. Again

resembling 2012, X22BL is significant for only one party (Right Cause) in the set D5>60, and

X22BL is significant for all candidates in ND5>60 and all but two parties (Just Russia and

KPRF) in ND5<60.16 j for United Russia is significantly greater in ND5<60 than in

ND5>60. In the sets D5>60, ND5>60 and ND5<60, only KPRF and United Russia have j > j

significantly. Neither of these parties were close to the threshold of seven percent needed to

gain a seat in the Duma,17 so there is no obvious basis in voters’ strategies that might be

related to the thresholds to explain this result.


The last-digit test in 2011 has three key similarities with 2012. X2LU∗ is never significant

for United Russia in any of the subsets, although the statistic in the set of all UIKs comes

close with αLU∗ = .07 (Table 5). In all sets X2LU < X2

LU∗ for United Russia. Ignoring the

cases where NL∗ < 100,18 X2LU∗ is not significant in D5>60 for any party except KPRF.19

Differences from 2012 are that X2LU∗ is significant for only one party—KPRF—in ND5>60,

and X2LU∗ is significant for only one party—-LDPR—in ND5<60. The similarities between

2011 and 2012 are the unexceptioned uniformity of the last digits in the United Russia vote

counts, the unique significance in D5>60 of X2LU∗ for KPRF, and the unusual distribution

observed for single parties in the subsets which mostly test free of vote augmentation in the

randomization tests. In 2011 the unusual results are that KPRF has a significant X2LU∗

value in ND5>60 and LDPR has a significant X2LU∗ value in ND5<60, while in 2012 the

unusual result was the insignificant X2LU∗ value for Zyuganov in ND5<60.

16The p-value that equals .03 is not significant if FDR adjustment is applied to the set of all p-values inD5>60, ND5>60 and ND5<60.

17Based on Central Election Commission of the Russian Federation (2013), United Russia received32,371,737 votes, which is 49.3 percent, while the other parties (Just Russia, LDPR, Patriots, KPRF,Yabloko, Right Cause) respectively received 13.2, 11.7, .97, 19.2, 3.4 and .60 percent.

18While it may be reasonable even with fewer cases to assume the distribution of X2LU∗

under the null isthe chi-squared distribution (Larntz 1978; Koehler and Larntz 1980; McCullagh 1986), in this case I avoidtaking that assumption seriously. The exact distributions could be computed when NL∗ is small. I haven’tdone that because not much of substance seems to be at stake with the very small parties.

19The p-value that equals .01 is not significant if FDR adjustment is applied to the set of all p-values inD5>60, ND5>60 and ND5<60.

17


The patterns in the digit tests in 2008 also resemble those in 2012, with one interesting

difference. The second-digit tests show that in the set of all UIKs X22BL is significant for all

candidates except Zyuganov (Table 6). In this set, j differs significantly from j for all

candidates. X22BL is significant for only one of the candidates (Zhirinovskij) in the set

D5>60, but X22BL is not significant for only a single candidate in both ND5>60 (Zyuganov)

and ND5<60 (Bogdanov). Again an important question is why there is such a large

difference between D5>60 and ND5>60 in X22BL for Medvedev and Bogdanov. In both sets

Medvedev received pretty much the same range of percentages of the votes. Perhaps the

difference comes down to sample size. j for Medvedev does not differ significantly between

the two sets, and neither does j for Bogdanov. Perhaps the second digits in D5>60 are not

2BL-distributed, but the sample size there—which is 24 or 35 times smaller than the

sample in ND5>60—is not big enough to trigger a significant X22BL value. For Medvedev

j < j in D5>60 and ND5>60 but j > j in ND5<60, a pattern that echoes the pattern

observed across the same sets for Putin in 2012.


The interesting variation in the last digit test results for 2008 is that the test suggests

that in at least one subset of the data the last digit distribution for the United Russia

candidate is not uniform. Table 7 shows that X2LU∗ is significant for Medvedev both in the

set of all UIKs and in D5>60.20 X2

LU∗ is not significant for Medvedev in ND5>60 or ND5<60.

X2LU∗ > X2

LU in three of the sets of UIKs but not in ND5<60. Ignoring the cases where

NL∗ < 100, X2LU∗ is significant for Zyuganov and Zhirinovskij in the set of all the UIKs and

in ND5>60 but not in D5>60 or ND5<60. Why the last digits of the vote counts for

Zyuganov and Zhirinovskij are uniform in UIKs where the randomization tests show their

votes are augmented (D5>60) but not in the UIKs where they are not (ND5>60) is as much

20The two p-value that equal .03 (one in D5>60 and one in ND5>60) are not significant if FDR adjustmentis applied to the set of all p-values in D5>60, ND5>60 and ND5<60.

18

a mystery as it is when the same pattern occurs in other years. Given all the strange

patterns in the tests, it hardly seems reasonable to conclude that the last digit tests show

that votes for Medvedev in 2008 were manipulated while the votes for United Russia in

2011 or for Putin in 2012 were not.


The digit tests for 2004 data suggest that the distinctive features observed in 2008 are

more about Medvedev (or about Putin) than about any trend over time. With regard to

Putin, at least, last digit tests in 2004 resemble those in 2012: X2LU∗ is never significant for

Putin in any of the subsets (Table 8). In the set of all UIKs X2LU∗ is significant for all

candidates that have NL∗ > 100 (including Against All), just as in 2012. In other respects

the last digit tests for 2004 resemble those in 2012 only slightly. Ignoring the cases where

NL∗ < 100, in D5>60 X2LU∗ is not significant for both of the other candidates while in 2012

X2LU∗ is significant for one out of three. In ND5>60 X2

LU∗ is not significant for two of the

four other candidates (including Against All) whereas in 2012 it is significant for all four

candidates. In ND5<60 X2LU∗ is not significant for any of the four candidates (including

Against All) while in 2012 X2LU∗ is significant for all but one of the four candidates. One

can speak of a trend across presidential elections, perhaps, in which last digits of vote

counts follow a uniform distribution with decreasing frequency as time progresses, but any

such trend does not apply to the vote counts for Putin. The significant last digit test

results for Medvedev are unique among the United Russia candidates.


Second-digit test statistics for 2004 somewhat resemble the results for 2012. In 2004

X22BL is always significant in the set of all UIKs and in ND5>60 (Table 9). In 2012 in these

two sets only X22BL for Zyuganov in the set of all UIKs is significant. In D5>60 X2

2BL is not

significant for five of the seven candidates (including Against All), whereas in 2012 in this

19

set X22BL is not significant for all five candidates.21 Despite the differences in X2

2BL between

D5>60 and ND5>60, j for each candidate does not differ significantly between the sets.

Perhaps, as in 2008, the different results for X22BL come down to the sample sizes: the

second digits in D5>60 may not be 2BL-distributed, and the sample size there is too small

to detect the distribution difference. In ND5<60 X22BL is not significant for four out of seven

candidates, which compares to X22BL being significant in this set for all five candidates in

2012. In this set j > j significantly for Putin and Haritonov despite the fact that X22BL is

not significant. This 2004 value of j for Putin is the same as j for Putin, Medvedev or

United Russia in this same set in the other elections.


4 Discussion

Simply referring to the digit tests in the most straightforward and unconditional way, they

produce results that hardly seem plausible. In light of the extensive evidence to the

contrary, it is not possible to believe what the last digit tests in the set of all UIKs in each

year would suggest, which is that counts of votes for Putin or for United Russia were not

fraudulently manipulated while, uniquely among the set of choices affiliated with United

Russia, the counts of votes for Medevev were fraudulently manipulated. The tests of

second digits in the set of all UIKs would trigger a diagnosis of fraud for Putin, Medvedev

and United Russia if interpreted as Pericchi and Torres (2011) would suggest. But even if

we do not acknowledge the doubts about the unconditional second-digit tests discussed in

Mebane (2013), in 2012 X22BL says that the vote counts for Putin do not follow 2BL while j

gives no reason to doubt that they do.

Unconditional digit tests suggest interesting conclusions about other candidates and

parties, but one wonders how credible are these suggestions. Do Zyuganov and KPRF

21In 2004, the p-value that equals .03 is not significant if FDR adjustment is applied to the set of allp-values in D5>60, ND5>60 and ND5<60.

20

uniquely among the alternatives in 2008 and 2011 have unmanipulated vote counts? That

is what both second-digit and last-digit tests suggest, moreover the second-digit tests

suggest a similar conclusion for 2012.

The randomization tests suggest vote counts for many candidates are augmented in

some UIKs relative to other “neighboring” UIKs, but introducing some conditioning in the

digit tests by separating the data into sets based on the randomization test results

produces a confusing picture. In every year, 2004–2011, both second-digit and last-digit

tests are less likely to be triggered in the set of UIKs in which there is demonstrably more

vote count augmentation (the set D5>60) than in the set of those UIKs’ “neighbors” (the

set ND5>60). It may be that the differences in test outcomes between the two sets are

artifacts of their respective sample sizes: the latter sets tend to be at least ten times larger

than the former sets. But the sets D5>60 are often not all that small. The least one can say

is that the results illuminate how little statistical power the chi-squared statistics have.

Regarding the last-digit tests, perhaps a prescription emerges from using them with the

data from Russia. Perhaps the idea should be not to use the test to try to isolate which

vote statistics are good and which are not, but instead to adopt a rule that any deviation

from a uniform distribution for any candidate should discredit the entire election. Such a

rule could be extended to say that fiding a significant deviation in any reasonable subset of

large-enough precincts for any candidate should discredit the election, given suitable

adjustments for multiple testing.

In every year except 2004, the second-digit test statistics have X22BL significant and

j ≈ 4.35 for United Russia in the set of UIKs (ND5<60) that have turnout and United

Russia (or Putin) vote percentage not divisible by five and vote percentages for United

Russia less than 60 percent. In ND5<60 in 2004 the value of j for Putin is similar even

though X22BL is not significant. These findings contrast with the results from UIKs in

which United Russia’s vote percentage is greater than 60 percent. In the sets of those

UIKs, j ≤ j. It is tempting to say that these values of j reflect there being smaller effects

21

of fraud among the vote counts that give Putin or United Russia smaller shares of the vote,

because having less fraud there allows evidence of votes being strategically added to

Putin’s votes to come through. A strong argument against this interpretation is that,

because the other candidates receive significant numbers of votes, it is difficult to know

what strategy the voters here might be using.

Is it plausible that votes have been concocted with an eye to passing the digit tests?

It’s certainly plausible that random number generators or other algorithms were used to

adjust vote counts, rather than relying on the numbers individual humans spontaneously

think up. If vote frauds are coordinated across all of Russia, whether through the signaling

scheme described by Kalinin and Mebane (2011) or not, and the assigned or chosen targets

are particular turnout or Putin/United Russia vote percentages and not particular totals

for the vote counts, then the vote counts that result might be uniformly distributed even

though many of them are faked. Whether such a mechanism can produce vote counts that

pass the second-digit tests according to X22BL is a question.

The results for 2004 pose the sharpest challenge to the preceding speculations. Neither

Pericchi and Torres (2004) nor even the conference draft versions of Beber and Scacco

(2012) existed at the time of the 2004 election. The election was in the month of May and

Pericchi and Torres (2004) did not appear until October of that year. The conference-paper

version of Beber and Scacco (2012) did not exist until at least two years later. It’s of

course wonderful to think that social science and statistical research is taken so seriously

that malefactors in one of the world’s superpowers actively respond to it. If that idea is to

be granted, then we can perhaps cite the forerunners of Pericchi and Torres (2004) and

Beber and Scacco (2012) that did exist in time to warn the alleged perpetrators. As

Pericchi and Torres (2004) and Beber and Scacco (2012) respectively acknowledge, Nigrini

(1996) suggested using Benford’s Law to detect tax cheating and Mosimann, Wiseman and

Edelman (1995) suggested checking the last digits to detect research fraud. And for the

general idea to deploy Benford’s Law against fraud, Varian (1972) comes even earlier. None

22

of the precursors had votes or specifically 2BL in mind, but there they were.

23

Figure 1: Turnout Divisible-by-five Tests: Russian 2012 Presidential Election

20 40 60 80 100

0.0

0.4

0.8

(a) ZhirinovskijLDPR

Turnout

p−va

lue

20 40 60 80 100

0.0

0.4

0.8

(b) ZyuganovKPRF

Turnout

p−va

lue

20 40 60 80 100

0.0

0.4

0.8

(c) MironovJust Russia

Turnout

p−va

lue

20 40 60 80 100

0.0

0.4

0.8

(d) ProhorovIndependent

Turnout

p−va

lue

20 40 60 80 1000.

00.

40.

8

(e) PutinUnited Russia

Turnout

p−va

lue

Note: ◦, randomization test p-value; •, p-value that is significant at test level α = .05 givenFDR correction across all tests for all candidates shown; dotted line locates the value .05.Party abbreviation legend: LDPR, Liberal Democratic Party of Russia; KPRF, CommunistParty of the Russian Federation.

24

Figure 2: Putin Vote Percentage Divisible-by-five Tests: Russian 2012 Presidential Election

20 40 60 80 100

0.0

0.4

0.8

(a) ZhirinovskijLDPR

Putin percentage

p−va

lue

20 40 60 80 100

0.0

0.4

0.8

(b) ZyuganovKPRF

Putin percentage

p−va

lue

20 40 60 80 100

0.0

0.4

0.8

(c) MironovJust Russia

Putin percentage

p−va

lue

20 40 60 80 100

0.0

0.4

0.8

(d) ProhorovIndependent

Putin percentage

p−va

lue

20 40 60 80 1000.

00.

40.

8

(e) PutinUnited Russia

Putin percentage

p−va

lue


25

Figure 3: Joint Distribution of Rounded Turnout and Putin Vote Percentage, 2012 Elections

turnout

35404550556065707580859095100

Putin percentage

3540

45 50556065707580859095100

count

050

100150200250300350400450

500

Note: joint density of UIK-level rounded turnout and rounded United Russia votepercentage.

26

Table 1: Divisible-by-five Local Peaks: Russian 2012 Presidential Election

Putin Putinturnout percentage turnout percentage

50 55 85 9550 70 90 8560 80 95 8565 65 100 2570 75 100 4070 80 100 4580 80 100 5085 80 100 5585 85 100 6085 90 100 100

Note: combinations of rounded turnout and Putin vote percentage values that occur morefrequently than combinations of turnout and vote percentage that differ by two or less ineach value; for instance, the pair (50, 55) occurs more frequently than any of the pairs{(i, j) : i = 48, 49, 51, 52, j = 53, 54, 56, 57}.

27

Figure 4: Turnout and Putin Vote Proportion Density, 2012 Election

turnout

0.4

0.6

0.8

Putin proportion

0.4

0.6

0.8

0

50

100

150

200

Note: smoothed joint density of UIK-level turnout and United Russia vote proportion.

28

Figure 5: Turnout and Medvedev Vote Proportion Density, 2008 Election

turnout

0.4

0.6

0.8

Medvedev proportion

0.4

0.6

0.8

0

100

200

300

Note: smoothed joint density of UIK-level turnout and Medvedev vote proportion.

29

Figure 6: Turnout and Putin Vote Proportion Density, 2004 Election

turnout

0.4

0.6

0.8

Putin proportion

0.4

0.6

0.8

0

100

200

300

400

500

Note: smoothed joint density of UIK-level turnout and Putin vote proportion.

30


20 40 60 80 100

0.0

0.4

0.8

(a) MedvedevUnited Russia

Turnout

p−va

lue

20 40 60 80 100

0.0

0.4

0.8

(b) ZhirinovskijLDPR

Turnout

p−va

lue

20 40 60 80 100

0.0

0.4

0.8

(c) ZyuganovKPRF

Turnout

p−va

lue

20 40 60 80 100

0.0

0.4

0.8

(d) BogdanovDemocratic

Turnout

p−va

lue


31

Figure 8: Medvedev Vote Percentage Divisible-by-five Tests: Russian 2008 Presidential Elec-tion

20 40 60 80 100

0.0

0.4

0.8

(a) MedvedevUnited Russia

Medvedev percentage

p−va

lue

20 40 60 80 100

0.0

0.4

0.8

(b) ZhirinovskijLDPR

Medvedev percentage

p−va

lue

20 40 60 80 100

0.0

0.4

0.8

(c) ZyuganovKPRF

Medvedev percentage

p−va

lue

20 40 60 80 100

0.0

0.4

0.8

(d) BogdanovDemocratic

Medvedev percentage

p−va

lue


32


20 40 60 80 100

0.0

0.4

0.8

(a) PutinIndependent

Turnout

p−va

lue

20 40 60 80 100

0.0

0.4

0.8

(b) HaritonovKPRF

Turnout

p−va

lue

20 40 60 80 100

0.0

0.4

0.8

(c) HakamadaIndependent

Turnout

p−va

lue

20 40 60 80 100

0.0

0.4

0.8

(d) MalyshkinLDPR

Turnout

p−va

lue

20 40 60 80 1000.

00.

40.

8

(e) MironovLife

Turnout

p−va

lue

20 40 60 80 100

0.0

0.4

0.8

(f) Galz’evIndependent

Turnout

p−va

lue

20 40 60 80 100

0.0

0.4

0.8

(g) Against All

Turnout

p−va

lue

Note: ◦, randomization test p-value; •, p-value that is significant at test level α = .05 givenFDR correction across all tests for all candidates shown; dotted line locates the value .05.Party abbreviation legend: LDPR, Liberal Democratic Party of Russia; KPRF, CommunistParty of the Russian Federation; Life, Russian Party of Life.

33

Figure 10: Putin Vote Percentage Divisible-by-five Tests: Russian 2004 Presidential Election

20 40 60 80 100

0.0

0.4

0.8

(a) PutinIndependent

Putin percentage

p−va

lue

20 40 60 80 100

0.0

0.4

0.8

(b) HaritonovKPRF

Putin percentage

p−va

lue

20 40 60 80 100

0.0

0.4

0.8

(c) HakamadaIndependent

Putin percentage

p−va

lue

20 40 60 80 100

0.0

0.4

0.8

(d) MalyshkinLDPR

Putin percentage

p−va

lue

20 40 60 80 1000.

00.

40.

8

(e) MironovLife

Putin percentage

p−va

lue

20 40 60 80 100

0.0

0.4

0.8

(f) Galz’evIndependent

Putin percentage

p−va

lue

20 40 60 80 100

0.0

0.4

0.8

(g) Against All

Putin percentage

p−va

lue

Note: ◦, randomization test p-value; •, p-value that is significant at test level α = .05 givenFDR correction across all tests for all candidates shown; dotted line locates the value .05.Party abbreviation legend: LDPR, Liberal Democratic Party of Russia; KPRF, CommunistParty of the Russian Federation; Life, Russian Party of Life.

34

Figure 11: Turnout and United Russia Vote Proportion Density, 2011 Election

turnout

0.4

0.6

0.8

United Russia proportion

0.40.6

0.8

0

100

200

300

400


35

Figure 12: Turnout and United Russia Vote Proportion Density, 2007 Election

turnout

0.4

0.6

0.8

United Russia proportion

0.4

0.6

0.8

0

100

200

300

400


36

Figure 13: Turnout Divisible-by-five Tests: Russian 2011 Duma Election

20 40 60 80 100

0.0

0.4

0.8

(a) Just Russia

Turnout

p−va

lue

20 40 60 80 100

0.0

0.4

0.8

(b) LDPR

Turnout

p−va

lue

20 40 60 80 100

0.0

0.4

0.8

(c) Patriots

Turnout

p−va

lue

20 40 60 80 100

0.0

0.4

0.8

(d) KPRF

Turnout

p−va

lue

20 40 60 80 1000.

00.

40.

8

(e) Yabloko

Turnout

p−va

lue

20 40 60 80 100

0.0

0.4

0.8

(f) United Russia

Turnout

p−va

lue

20 40 60 80 100

0.0

0.4

0.8

(g) Right Cause

Turnout

p−va

lue


37

Figure 14: United Russia Vote Percentage Divisible-by-five Randomization Tests: Russian2011 Duma Election

20 40 60 80 100

0.0

0.4

0.8

(a) Just Russia

United Russia percentage

p−va

lue

20 40 60 80 100

0.0

0.4

0.8

(b) LDPR


p−va

lue

20 40 60 80 100

0.0

0.4

0.8

(c) Patriots


p−va

lue

20 40 60 80 100

0.0

0.4

0.8

(d) KPRF


p−va

lue

20 40 60 80 100

0.0

0.4

0.8

(e) Yabloko


p−va

lue

20 40 60 80 100

0.0

0.4

0.8

(f) United Russia


p−va

lue

20 40 60 80 100

0.0

0.4

0.8

(g) Right Cause


p−va

lue


38

Figure 15: Turnout Divisible-by-five Tests: Russian 2007 Duma Election

20 40 60 80 100

0.0

0.4

0.8

(a) United Russia

Turnout

p−va

lue

20 40 60 80 100

0.0

0.4

0.8

(b) Just Russia

Turnout

p−va

lue

20 40 60 80 100

0.0

0.4

0.8

(c) Patriots

Turnout

p−va

lue

20 40 60 80 100

0.0

0.4

0.8

(d) SPS

Turnout

p−va

lue

20 40 60 80 1000.

00.

40.

8

(e) Civilian Power

Turnout

p−va

lue

20 40 60 80 100

0.0

0.4

0.8

(f) Agrarian

Turnout

p−va

lue

20 40 60 80 100

0.0

0.4

0.8

(g) Yabloko

Turnout

p−va

lue

20 40 60 80 100

0.0

0.4

0.8

(h) Social Justice

Turnout

p−va

lue

20 40 60 80 100

0.0

0.4

0.8

(i) KPRF

Turnoutp−

valu

e

20 40 60 80 100

0.0

0.4

0.8

(j) LDPR

Turnout

p−va

lue

20 40 60 80 100

0.0

0.4

0.8

(k) Democratic

Turnout

p−va

lue

Note: ◦, randomization test p-value; •, p-value that is significant at test level α = .05 givenFDR correction across all tests for all candidates shown; dotted line locates the value .05.Party abbreviation legend: LDPR, Liberal Democratic Party of Russia; KPRF, CommunistParty of the Russian Federation; SPS, Union of Rightist Forces.

39

Figure 16: United Russia Vote Percentage Divisible-by-five Randomization Tests: Russian2007 Duma Election

20 40 60 80 100

0.0

0.4

0.8

(a) United Russia


p−va

lue

20 40 60 80 100

0.0

0.4

0.8

(b) Just Russia


p−va

lue

20 40 60 80 100

0.0

0.4

0.8

(c) Patriots


p−va

lue

20 40 60 80 100

0.0

0.4

0.8

(d) SPS


p−va

lue

20 40 60 80 100

0.0

0.4

0.8

(e) Civilian Power


p−va

lue

20 40 60 80 100

0.0

0.4

0.8

(f) Agrarian


p−va

lue

20 40 60 80 100

0.0

0.4

0.8

(g) Yabloko


p−va

lue

20 40 60 80 100

0.0

0.4

0.8

(h) Social Justice


p−va

lue

20 40 60 80 100

0.0

0.4

0.8

(i) KPRF

United Russia percentagep−

valu

e

20 40 60 80 100

0.0

0.4

0.8

(j) LDPR


p−va

lue

20 40 60 80 100

0.0

0.4

0.8

(k) Democratic


p−va

lue

Note: ◦, randomization test p-value; •, p-value that is significant at test level α = .05 givenFDR correction across all tests for all candidates shown; dotted line locates the value .05.Party abbreviation legend: LDPR, Liberal Democratic Party of Russia; KPRF, CommunistParty of the Russian Federation; SPS, Union of Rightist Forces.

40

Figure 17: Turnout and United Russia Vote Proportion Density, 2003 Election, PR Votes

turnout 0.4

0.6

0.8

United Russia proportion0.40.6

0.8

0

50

100

150


41

Figure 18: Turnout Divisible-by-five Tests: Russian 2003 Duma Election, Proprotional Votes

20 40 60 80 100

0.0

0.4

0.8

(a) Against All

Turnout

p−va

lue

20 40 60 80 100

0.0

0.4

0.8

(b) United Russia

Turnout

p−va

lue

20 40 60 80 100

0.0

0.4

0.8

(c) KPRF

Turnout

p−va

lue

20 40 60 80 100

0.0

0.4

0.8

(d) LDPR

Turnout

p−va

lue

20 40 60 80 1000.

00.

40.

8

(e) Rodina

Turnout

p−va

lue

20 40 60 80 100

0.0

0.4

0.8

(f) Yabloko

Turnout

p−va

lue

20 40 60 80 100

0.0

0.4

0.8

(g) SPS

Turnout

p−va

lue

20 40 60 80 100

0.0

0.4

0.8

(h) Agrarian

Turnout

p−va

lue

20 40 60 80 100

0.0

0.4

0.8

(i) Pensioners

Turnoutp−

valu

e

20 40 60 80 100

0.0

0.4

0.8

(j) Rebirth

Turnout

p−va

lue

20 40 60 80 100

0.0

0.4

0.8

(k) People’s

Turnout

p−va

lue

Note: ◦, randomization test p-value; •, p-value that is significant at test level α = .05 givenFDR correction across all tests for all candidates shown; dotted line locates the value .05.Party abbreviation legend: LDPR, Liberal Democratic Party of Russia; KPRF, CommunistParty of the Russian Federation; SPS, Union of Rightist Forces; Rebirth, Party of Russia’sRebirth-Russian Party of Life; People’s, People’s Party of the Russian Federation.

42

Figure 19: United Russia Vote Percentage Divisible-by-five Randomization Tests: Russian2003 Duma Election, Proprotional Votes

20 40 60 80 100

0.0

0.4

0.8

(a) Against All


p−va

lue

20 40 60 80 100

0.0

0.4

0.8

(b) United Russia


p−va

lue

20 40 60 80 100

0.0

0.4

0.8

(c) KPRF


p−va

lue

20 40 60 80 100

0.0

0.4

0.8

(d) LDPR


p−va

lue

20 40 60 80 100

0.0

0.4

0.8

(e) Rodina


p−va

lue

20 40 60 80 100

0.0

0.4

0.8

(f) Yabloko


p−va

lue

20 40 60 80 100

0.0

0.4

0.8

(g) SPS


p−va

lue

20 40 60 80 100

0.0

0.4

0.8

(h) Agrarian


p−va

lue

20 40 60 80 100

0.0

0.4

0.8

(i) Pensioners


p−va

lue

20 40 60 80 100

0.0

0.4

0.8

(j) Rebirth


p−va

lue

20 40 60 80 100

0.0

0.4

0.8

(k) People’s


p−va

lue

Note: ◦, randomization test p-value; •, p-value that is significant at test level α = .05 givenFDR correction across all tests for all candidates shown; dotted line locates the value .05.Party abbreviation legend: LDPR, Liberal Democratic Party of Russia; KPRF, CommunistParty of the Russian Federation; SPS, Union of Rightist Forces; Rebirth, Party of Russia’sRebirth-Russian Party of Life; People’s, People’s Party of the Russian Federation.

43

Figure 20: Turnout and United Russia Vote Proportion Density, 2003 Election, PluralityVotes

turnout 0.4

0.6

0.8

United Russia proportion0.40.6

0.8

0

100

200

300


44

Figure 21: Turnout Divisible-by-five Tests: Russian 2003 Duma Election, Plurality Votes

20 40 60 80 100

0.0

0.4

0.8

(a) Against All

Turnout

p−va

lue

20 40 60 80 100

0.0

0.4

0.8

(b) None

Turnout

p−va

lue

20 40 60 80 100

0.0

0.4

0.8

(c) United Russia

Turnout

p−va

lue

20 40 60 80 100

0.0

0.4

0.8

(d) KPRF

Turnout

p−va

lue

20 40 60 80 1000.

00.

40.

8

(e) LDPR

Turnout

p−va

lue

20 40 60 80 100

0.0

0.4

0.8

(f) Rodina

Turnout

p−va

lue

20 40 60 80 100

0.0

0.4

0.8

(g) Yabloko

Turnout

p−va

lue

20 40 60 80 100

0.0

0.4

0.8

(h) Agrarian

Turnout

p−va

lue

20 40 60 80 100

0.0

0.4

0.8

(i) Pensioners

Turnoutp−

valu

e

20 40 60 80 100

0.0

0.4

0.8

(j) Rebirth

Turnout

p−va

lue

20 40 60 80 100

0.0

0.4

0.8

(k) People’s

Turnout

p−va

lue

20 40 60 80 100

0.0

0.4

0.8

(l) Independent

Turnout

p−va

lue

Note: ◦, randomization test p-value; •, p-value that is significant at test level α = .05 givenFDR correction across all tests for all candidates shown plus 204 tests for other parties;dotted line locates the value .05.Party abbreviation legend: LDPR, Liberal Democratic Party of Russia; KPRF, CommunistParty of the Russian Federation; SPS, Union of Rightist Forces; Rebirth, Party of Russia’sRebirth-Russian Party of Life; People’s, People’s Party of the Russian Federation.

45

Figure 22: Putin Vote Percentage Divisible-by-five Tests: Russian 2003 Duma Election,Plurality Votes

20 40 60 80 100

0.0

0.4

0.8

(a) Against All


p−va

lue

20 40 60 80 1000.

00.

40.

8

(b) None


p−va

lue

20 40 60 80 100

0.0

0.4

0.8

(c) United Russia


p−va

lue

20 40 60 80 100

0.0

0.4

0.8

(d) KPRF


p−va

lue

20 40 60 80 100

0.0

0.4

0.8

(e) LDPR


p−va

lue

20 40 60 80 100

0.0

0.4

0.8

(f) Rodina


p−va

lue

20 40 60 80 100

0.0

0.4

0.8

(g) Yabloko


p−va

lue

20 40 60 80 100

0.0

0.4

0.8

(h) Agrarian


p−va

lue

20 40 60 80 100

0.0

0.4

0.8

(i) Pensioners


p−va

lue

20 40 60 80 100

0.0

0.4

0.8

(j) Rebirth


p−va

lue

20 40 60 80 100

0.0

0.4

0.8

(k) People’s


p−va

lue

20 40 60 80 100

0.0

0.4

0.8

(l) Independent


p−va

lue

Note: ◦, randomization test p-value; •, p-value that is significant at test level α = .05 givenFDR correction across all tests for all candidates shown plus 169 tests for other parties;dotted line locates the value .05.Party abbreviation legend: LDPR, Liberal Democratic Party of Russia; KPRF, CommunistParty of the Russian Federation; SPS, Union of Rightist Forces; Rebirth, Party of Russia’sRebirth-Russian Party of Life; People’s, People’s Party of the Russian Federation.

46

Table 2: Second-digit Tests, Russian Presidential Elections, 2012

Set Party N χ22BL α j jlo jhi

All Zhirinovskij 74,808 608.80 .00 3.96 3.94 3.98Zyuganov 86,742 13.67 .42 4.19 4.17 4.21Mironov 60,290 52.61 .00 4.13 4.10 4.15Prohorov 63,895 275.53 .00 4.00 3.98 4.02Putin 94,937 67.47 .00 4.21 4.19 4.22

D5>60a Zhirinovskij 1,724 7.35 .45 4.06 3.92 4.19

Zyuganov 2,198 4.70 .26 4.12 3.99 4.24Mironov 1,178 12.63 .46 3.99 3.82 4.15Prohorov 1,314 17.38 .27 3.97 3.81 4.14Putin 3,119 9.55 .50 4.08 3.98 4.18

ND5>60b Zhirinovskij 25,861 71.16 .00 4.06 4.03 4.10


ND5<60c Zhirinovskij 20,933 481.69 .00 3.81 3.77 3.86


Notes: a rounded turnout and rounded Putin vote percentage divisible by 5 and Putin votepercentage > 60. b neither rounded turnout nor rounded Putin vote percentage divisible by5 and rounded Putin vote percentage > 60. c neither rounded turnout nor rounded Putinvote percentage divisible by 5 and rounded Putin vote percentage ≤ 60. Statistics for UIKvote counts. N is the number of UIKs with a vote count > 9. α = (1 + [−ep log(p)]−1)−1

where p is the p-value of X22BL. jlo and jhi are the lower and upper bounds of the 95%

confidence interval for j.Data source: .

47

Table 3: Last-digit Tests, Russian Presidential Elections, 2012

Set Party χ2LU αLU χ2

LU∗ αLU∗ NL∗

All Zhirinovskij 1007.16 .00 32.47 .00 11,507Zyuganov 156.70 .00 36.68 .00 47,090Mironov 2979.84 .00 61.16 .00 1,899Prohorov 2607.31 .00 43.23 .00 21,097Putin 13.48 .43 23.75 .06 83,047

D5>60a Zhirinovskij 1010.64 .00 14.89 .38 122

Zyuganov 658.90 .00 27.94 .02 700Mironov 1087.72 .00 20.82 .14 11Prohorov 1250.80 .00 11.32 .49 118Putin 5.79 .36 6.58 .42 2,597

ND5>60b Zhirinovskij 452.23 .00 37.05 .00 2,074

Zyuganov 49.10 .00 49.91 .00 11,752Mironov 1790.97 .00 24.56 .05 225Prohorov 1444.44 .00 24.35 .05 2,655Putin 4.90 .28 12.34 .46 31,018

ND5<60c Zhirinovskij 11.48 .48 79.19 .00 5,249

Zyuganov 4.02 .19 5.00 .29 17,939Mironov 32.72 .00 40.53 .00 984Prohorov 32.30 .00 26.09 .03 10,840Putin 8.52 .49 8.71 .49 20,141

Notes: a rounded turnout and rounded Putin vote percentage divisible by 5 and Putin votepercentage > 60. b neither rounded turnout nor rounded Putin vote percentage divisible by5 and rounded Putin vote percentage > 60. c neither rounded turnout nor rounded Putinvote percentage divisible by 5 and rounded Putin vote percentage ≤ 60. Statistics for UIKvote counts. αk = (1 + [−epk log(pk)]

−1)−1 where pk is the p-value of X2k , k ∈ {LU, LU∗}.

NL: All, 95,414; D5>60, 3,138; ND5>60, 36,257; ND5<60, 21,960.Data source: .

48

Table 4: Second-digit Digit Tests, Russian Duma Elections, 2011


All Just Russia 79,165 65.54 .00 4.11 4.09 4.13LDPR 79,539 181.59 .00 4.06 4.04 4.08Patriots 25,943 1260.31 .00 3.56 3.52 3.59KPRF 86,282 20.92 .13 4.19 4.17 4.21Yabloko 41,499 108.16 .00 4.05 4.02 4.08United Russia 94,575 35.02 .00 4.24 4.22 4.26Right Cause 12,982 2225.36 .00 3.02 2.97 3.06

D5>60a Just Russia 793 19.58 .18 3.90 3.69 4.11

LDPR 782 15.93 .33 3.80 3.60 4.01Patriots 99 20.46 .15 3.38 2.82 3.94KPRF 1,105 15.69 .34 4.01 3.83 4.19Yabloko 205 22.01 .10 3.43 3.04 3.83United Russia 1,826 21.70 .11 4.07 3.94 4.20Right Cause 64 32.44 .00 2.39 1.76 3.02

ND5>60b Just Russia 9,449 104.50 .00 3.90 3.84 3.96

LDPR 9,374 88.73 .00 3.95 3.89 4.01Patriots 1,205 127.41 .00 3.26 3.11 3.42KPRF 12,755 47.88 .00 4.07 4.02 4.12Yabloko 2,216 76.86 .00 3.71 3.59 3.83United Russia 15,964 26.02 .03 4.10 4.05 4.14Right Cause 580 97.14 .00 3.09 2.87 3.31

ND5<60c Just Russia 39,817 13.69 .42 4.16 4.13 4.19

LDPR 40,132 77.45 .00 4.09 4.07 4.12Patriots 15,143 713.68 .00 3.57 3.52 3.61KPRF 40,785 23.72 .06 4.25 4.22 4.28Yabloko 23,813 54.68 .00 4.06 4.03 4.10United Russia 41,624 80.94 .00 4.30 4.27 4.33Right Cause 7,542 1240.75 .00 3.03 2.97 3.09




49

Table 5: Last-digit Tests, Russian Duma Elections, 2011


LU∗ αLU∗ NL∗

All Just Russia 1140.98 .00 43.13 .00 35,730LDPR 1574.35 .00 53.64 .00 33,714Patriots 30168.79 .00 6.70 .42 27KPRF 202.03 .00 59.02 .00 46,404Yabloko 27166.87 .00 28.24 .02 5,132United Russia 14.57 .39 23.34 .07 76,835Right Cause 67106.01 .00 4.33 .22 15

D5>60a Just Russia 1097.72 .00 16.92 .29 123

LDPR 1410.80 .00 10.27 .50 132Patriots 4361.95 .00 9.00 .50 1KPRF 564.67 .00 29.96 .01 394Yabloko 3945.29 .00 8.71 .49 7United Russia 5.53 .34 12.22 .47 1,516Right Cause 5562.26 .00 — — 0

ND5>60b Just Russia 1348.39 .00 6.55 .41 1,432

LDPR 2014.72 .00 18.45 .22 1,311Patriots 21248.80 .00 9.00 .50 1KPRF 95.08 .00 53.70 .00 3,625Yabloko 21146.29 .00 16.71 .30 62United Russia 10.78 .49 18.33 .23 13,545Right Cause 35240.75 .00 8.00 .48 2

ND5<60c Just Russia 9.15 .50 23.67 .07 21,154

LDPR 17.23 .28 38.24 .00 19,976Patriots 4023.05 .00 11.00 .49 15KPRF 7.92 .47 21.23 .12 25,480Yabloko 2983.74 .00 17.81 .25 3,132United Russia 8.50 .49 14.96 .37 33,529Right Cause 12925.08 .00 5.00 .29 5




50



All Medvedev 95,883 124.22 .00 4.11 4.09 4.13Zyuganov 87,298 18.48 .22 4.22 4.20 4.24Bogdanov 35,308 212.55 .00 3.97 3.94 4.00Zhirinovskij 79,650 434.69 .00 3.98 3.96 4.00

D5>60a Medvedev 3,864 18.95 .20 4.04 3.95 4.14

Zyuganov 3,025 13.20 .44 4.09 3.98 4.19Bogdanov 1,020 20.20 .16 3.81 3.63 3.98Zhirinovskij 2,638 45.48 .00 3.82 3.71 3.93

ND5>60b Medvedev 92,019 115.61 .00 4.11 4.09 4.13


ND5<60c Medvedev 10,448 38.36 .00 4.27 4.21 4.33




confidence interval for j. Number of UIKs used for χ2LU : All, 95,414; D5>60, 3,138; ND5>60,

36,257; ND5<60, 21,960.Data source: .

51



LU∗ αLU∗ NL∗

All Medvedev 27.27 .02 37.46 .00 85,259Zyuganov 120.58 .00 29.57 .01 48,095Bogdanov 24268.18 .00 9.33 .50 84Zhirinovskij 727.71 .00 69.70 .00 29,479

D5>60a Medvedev 25.40 .04 26.44 .03 3,289

Zyuganov 371.80 .00 11.39 .49 1,422Bogdanov 2685.60 .00 8.00 .48 2Zhirinovskij 605.96 .00 9.82 .50 808

ND5>60b Medvedev 20.69 .14 21.03 .13 42,926

Zyuganov 83.07 .00 25.89 .03 21,874Bogdanov 22134.04 .00 7.71 .47 28Zhirinovskij 542.12 .00 45.96 .00 13,260

ND5<60c Medvedev 10.35 .50 10.01 .50 8,867

Zyuganov 13.64 .42 14.37 .40 7,731Bogdanov 787.69 .00 12.81 .45 21Zhirinovskij 10.07 .50 16.49 .31 5,144




52



LU∗ αLU∗ NL∗

All Putin 14.99 .37 21.67 .11 83,603Haritonov 394.37 .00 29.63 .01 39,011Hakamada 8108.85 .00 46.51 .00 5,508Malyshkin 6063.29 .00 13.75 .42 67Mironov 27563.77 .00 15.46 .35 13Glaz’ev 6935.32 .00 53.25 .00 4,346Against All 9663.77 .00 46.61 .00 2,565

D5>60a Putin 13.47 .43 18.64 .22 3,547

Haritonov 1235.88 .00 16.48 .31 1,101Hakamada 2671.71 .00 14.14 .41 169Malyshkin 2925.17 .00 — — 0Mironov 4632.69 .00 9.00 .50 1Glaz’ev 2692.00 .00 12.06 .47 85Against All 2680.84 .00 13.12 .44 82

ND5>60b Putin 12.47 .46 18.86 .21 44,380

Haritonov 27.43 .02 19.69 .18 18,898Hakamada 3491.45 .00 38.44 .00 2,883Malyshkin 2185.61 .00 14.18 .40 17Mironov 11832.53 .00 20.14 .16 7Glaz’ev 2580.90 .00 42.59 .00 1,695Against All 4101.41 .00 15.28 .36 1,141

ND5<60c Putin 7.01 .44 8.62 .49 7,116

Haritonov 2.55 .05 9.91 .50 5,908Hakamada 264.01 .00 2.72 .06 633Malyshkin 78.94 .00 8.67 .49 18Mironov 1659.16 .00 8.00 .48 2Glaz’ev 173.18 .00 8.36 .49 1,109Against All 408.32 .00 15.43 .35 460




53

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All Putin 95,128 292.03 0 4.10 4.08 4.12Haritonov 84,626 139.59 0 4.08 4.06 4.09Hakamada 50,087 243.15 0 4.01 3.99 4.04Malyshkin 52,131 162.21 0 4.04 4.01 4.06Mironov 18,963 2854.57 0 3.08 3.05 3.12Glaz’ev 54,879 154.59 0 4.07 4.04 4.09Against All 50,627 109.93 0 4.11 4.08 4.13

D5>60a Putin 4,267 16.52 .31 4.06 3.97 4.15

Haritonov 2,914 16.80 .29 4.06 3.95 4.16Hakamada 1,682 26.10 .03 3.97 3.83 4.11Malyshkin 1,682 18.18 .23 3.96 3.82 4.10Mironov 623 105.24 .00 3.09 2.88 3.30Glaz’ev 1,801 11.64 .48 4.12 3.98 4.26Against All 1,696 9.42 .50 4.09 3.95 4.23

ND5>60b Putin 48,824 205.21 0 4.09 4.06 4.12

Haritonov 44,188 120.88 0 4.04 4.01 4.07Hakamada 26,541 160.32 0 3.99 3.95 4.02Malyshkin 27,069 90.73 0 4.03 4.00 4.07Mironov 10,045 1483.83 0 3.10 3.05 3.15Glaz’ev 28,748 50.66 0 4.10 4.06 4.13Against All 26,594 40.44 0 4.13 4.10 4.17

ND5<60c Putin 8,909 14.00 .41 4.28 4.22 4.34

Haritonov 8,853 20.94 .13 4.29 4.23 4.36Hakamada 4,998 20.58 .14 4.09 4.01 4.17Malyshkin 5,801 20.34 .15 4.10 4.02 4.17Mironov 1,982 404.65 .00 2.93 2.81 3.04Glaz’ev 5,857 39.94 .00 4.01 3.94 4.09Against All 5,243 47.60 .00 4.00 3.92 4.08




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Documents

UsingVoteCounts’DigitstoDiagnoseStrategiesand Frauds: Russiawmebane/apsa13.pdfOver the most recent election cycles Russian elections have become increasingly unfree and unfair, characterized